GMAT Quant Preparation: Prime and Composite Numbers

Goalisb
Jul 11
7 min read

We've explored how numbers relate through division, how they grow through multiples, and how their parity (even/odd) behaves. Now, let's zoom in on the most fundamental types of integers: prime numbers and composite numbers. Think of prime numbers as the "atomic elements" of the number system – they cannot be broken down further by multiplication. Composite numbers are like "molecules" made from these prime "atoms."

Understanding primes is not just an academic exercise; it's absolutely vital for mastering concepts like prime factorization, finding common denominators, simplifying fractions, and tackling various number properties problems on the GMAT. This lesson will provide a deep dive into these essential classifications and their applications.

Learning Objectives:

By the end of this lesson, you will be able to:

Clearly and accurately define prime and composite numbers.
Understand the unique status of the number 1.
Master the process of prime factorization for any composite number.
Utilize prime factorization as a foundational skill for future number theory topics.
Identify common misconceptions about prime and composite numbers.

A. Prime Numbers: The Indivisible Atoms

Imagine you have a certain number of identical square tiles, and you want to arrange them into a perfect rectangle (not just a single row). If you have 7 tiles, the only way you can arrange them into a rectangle is a 1 x 7 row (or a 7 x 1 column). You can't make a 2 x X or 3 x Y rectangle. This unique property means 7 is a prime number.

Formal Definition: A prime number is a positive integer greater than 1 that has exactly two distinct positive factors (divisors):

The number 1
The number itself

Examples of Prime Numbers:

2: Factors are {1, 2}. (It's the only even prime number!)
3: Factors are {1, 3}.
5: Factors are {1, 5}.
7: Factors are {1, 7}.
11: Factors are {1, 11}.
13: Factors are {1, 13}.
...and so on.

Key Characteristics of Prime Numbers:

Smallest Prime: 2 is the smallest prime number.
Only Even Prime: 2 is the only even prime number. All other prime numbers are odd. If a number greater than 2 is even, it's automatically divisible by 2 (and 1 and itself), meaning it has at least 3 factors, making it composite.
Building Blocks: Prime numbers are the fundamental building blocks for all other integers (except 0 and 1) through multiplication.

B. Composite Numbers: The Molecular Structures

Continuing our tile analogy: If you have 12 tiles, you can arrange them in a 1 x 12 row, but also a 2 x 6 rectangle, or a 3 x 4 rectangle. Because you can form more than one unique rectangle (other than just a single row), 12 is a composite number.

Formal Definition: A composite number is a positive integer greater than 1 that has more than two distinct positive factors (divisors). In other words, it can be evenly divided by at least one number other than 1 and itself.

Examples of Composite Numbers:

4: Factors are {1, 2, 4}.
6: Factors are {1, 2, 3, 6}.
8: Factors are {1, 2, 4, 8}.
9: Factors are {1, 3, 9}. (Often mistakenly thought to be prime!)
10: Factors are {1, 2, 5, 10}.
12: Factors are {1, 2, 3, 4, 6, 12}.
...and so on.

C. The Unique Case: The Number 1

The number 1 is a special case. It fits neither the definition of a prime number nor a composite number.

Why not prime? A prime number must have exactly two distinct positive factors. The number 1 only has one positive factor: itself (1).
Why not composite? A composite number must have more than two distinct positive factors. The number 1 only has one.

Therefore, 1 is neither prime nor composite. This is a crucial distinction for the GMAT.

D. Prime Factorization: Deconstructing Numbers into Their Primes

Every composite number can be uniquely expressed as a product of prime numbers. This process is called prime factorization. It's like finding the unique "recipe" of prime ingredients for any number.

The Fundamental Theorem of Arithmetic: This theorem states that every integer greater than 1 is either a prime number itself or can be uniquely represented as a product of prime numbers (ignoring the order of the factors). This uniqueness is what makes prime factorization so powerful.

Methods for Prime Factorization:

There are two common visual methods:

1. The Factor Tree Method: Start with the number and break it down into any two factors. Continue breaking down non-prime factors until all branches end in prime numbers.

Example: Prime factorize 72

        72
       /  \
      8    9    (Any two factors of 72)
     / \  / \
    2  4  3  3  (Break down 8 into 2*4, and 9 into 3*3. 2, 3 are prime)
       / \
      2   2     (Break down 4 into 2*2. All are prime now)

The prime factors are the numbers at the end of the "leaves" of the tree. So, the prime factorization of 72 is 2 2 2 3 3, or in exponential form: 2^3 * 3^2.

2. The Division Method (Ladder Method): Start by dividing the number by the smallest prime number that divides it evenly. Continue dividing the result by prime numbers until the quotient is 1.

Example: Prime factorize 100

2 | 100
--
2 | 50
--
5 | 25
--
5 | 5
--
  | 1

The prime factors are all the divisors on the left side of the ladder. So, the prime factorization of 100 is 2 2 5 5, or in exponential form: 2^2 5^2.

Why is Prime Factorization Important?

Finding LCM and GCD: It's the most reliable method for finding the Least Common Multiple (LCM) and Greatest Common Divisor (GCD) of two or more numbers (covered in the next lesson!).
Simplifying Fractions: Prime factors help you find common factors to reduce fractions to their simplest form.
Number Properties: Many GMAT questions about divisibility, number of factors, perfect squares/cubes, or finding unknown integers rely heavily on understanding their prime factorization.
Roots: If a number is a perfect square or perfect cube, its prime factors will have even or multiples-of-three exponents, respectively.

E. Tips for Identifying Prime Numbers (and avoiding common mistakes)

Check Divisibility by Small Primes: To check if a number is prime, you only need to try dividing it by prime numbers (2, 3, 5, 7, 11, etc.) up to its square root.
- Example: To check if 101 is prime:
  - √101 is approximately 10.05.
  - So, we only need to check prime numbers up to 10: 2, 3, 5, 7.
  - 101 is not divisible by 2 (odd).
  - 101 is not divisible by 3 (1+0+1 = 2, not div by 3).
  - 101 is not divisible by 5 (doesn't end in 0 or 5).
  - 101 ÷ 7 = 14 with a remainder of 3 (not div by 7).
  - Since it's not divisible by any prime up to its square root, 101 is a prime number.
Common Misconceptions:
- 1 is prime: No, 1 is neither prime nor composite.
- Any odd number is prime: No. Examples: 9 (33), 15 (35), 21 (37), 25 (55), 27 (3*9), etc., are all odd but composite.
- Large numbers must be prime: No. A number like 111 is 3 * 37, so it's composite.

Interactive Check Your Understanding:

Is 51 a prime or composite number? If composite, list its factors.
What is the smallest prime number?
Prime factorize the number 90. Write your answer in exponential form.
Why is 1 not considered a prime number?

Practice Questions:

Classify each of the following numbers as Prime, Composite, or Neither: a) 29 b) 1 c) 87 d) 2 e) 121
Find the prime factorization of 144 using either method. Write the answer in exponential form.
The prime factorization of an unknown number N is 2^2 3^1 5^3. What is the value of N?
If a number P is prime, and P > 2, what is the parity (even or odd) of P?
The number of distinct positive factors of an integer K is 3. What kind of number is K? (e.g., prime, composite, square, etc.) Give an example of such a number.

Solutions to Practice Questions:

Classify numbers: a) 29: Prime (only factors are 1, 29) b) 1: Neither (only one factor, 1) c) 87: Composite (factors include 1, 3, 29, 87; 87 = 3 29) d) 2: Prime (only factors are 1, 2; the smallest and only even prime) e) 121: Composite (factors include 1, 11, 121; 121 = 11 11)
Prime factorization of 144: Using factor tree or division method: 144 = 2 72 72 = 2 36 36 = 2 18 18 = 2 9 9 = 3 3 So, 144 = 2 2 2 2 3 3 In exponential form: 2^4 * 3^2.
Value of N from prime factorization 2^2 3^1 5^3: N = (2 2) (3) (5 5 5) N = 4 3 125 N = 12 125 N = 1,500 The value of N is 1,500.
If P is prime and P > 2, what is its parity? The only even prime number is 2. Any other even number (greater than 2) would be divisible by 2, and therefore have more than two factors (1, 2, and itself), making it composite. Therefore, if P is a prime number greater than 2, it must be Odd.
Number of distinct positive factors of K is 3. What kind of number is K? Give an example. Numbers with an odd number of factors are always perfect squares. For a number to have exactly 3 factors, it must be the square of a prime number. Let K = p^x (where p is prime). The number of factors is x+1. If x+1 = 3, then x = 2. So, K must be of the form p^2. Example:
- If p = 2, K = 2^2 = 4. Factors of 4 are {1, 2, 4} (3 factors).
- If p = 3, K = 3^2 = 9. Factors of 9 are {1, 3, 9} (3 factors).
- If p = 5, K = 5^2 = 25. Factors of 25 are {1, 5, 25} (3 factors). So, K is a perfect square that is the square of a prime number. An example is 9.